On Interpolation to a given Analytic Function by Analytic Functions of Minimum Norm

and let the function f(z) be analytic in these points. To study the convergence to f(z) of the sequence of functions gn(z); here gn(z) is analytic throughout Rx, coincides with f(z) in the points 0nl, 0n2, • • • , 0nn, and among all functions with these two properties has the least norm in Rx. This problem has been previously studied [6; 7] where norm is [lub |g„(z)|, z in A\], and is now to be studied (§1) where norm is measured by a surface integral over Rlt or (§2) a parametric integral over the boundary of Rx, or (§3) a line integral over the boundary of A\. If the norm is measured by the integral of the square of the modulus, we obtain by this method an expansion of f(z) in a series of orthogonal functions, an expansion whose convergence properties we study (§4) in some detail. The asymptotic behavior of these orthogonal functions themselves and of their zeros is investigated in §5. 1. Interpolation by functions of minimum norm, surface integrals. If Ai is a given region, we define jQ'(Rx) (0 °°) lub [| F(z)\, z in A\]. We define J&(Ri) as the subclass of Q(Ri) consisting of those functions of Jif(Rx) which coincide with the given f(z) in the points p\,i, 0n2, • • • , /8„„. The functions of class J^n(Ri) form a normal family in Rx, and standard methods show that there exists at least one such function Fn(z) of minimum norm. The function Fn(z) is unique if 1 <g< oo, and also if q = oo and Rx is simply connected. If 5 is any point set, we denote its closure by 5. With the generic notation